Willmore submanifolds of the Möbius space and a Bernstein-type theorem
type: Item
instance of: scholarly article
title: Willmore submanifolds of the Möbius space and a Bernstein-type theorem
zbMATH Open document ID: 0790.53049
DOI: 10.1007/BF02567854
author: M. Rigoli
author: Isabel Salavessa
published in: Manuscripta Mathematica
publication date: 1994-06-26T00:00:00Z
full work available at URL: https://eudml.org/doc/155897
review text: After calculating the structure equations and transformation formulae for Darboux frames of an immersion \(f:M^ m\to S^ n\) into the Möbius space \(S^ n\subset {\mathbb{R}}P^{n-1}\), the authors give a generalization of the Willmore functional for such immersions. Critical immersions of this functional are called ``Willmore submanifolds'', the Euler-Lagrange equation for the corresponding variational problem is derived. The hyperbolic conformal Gauß map, taking values in the conformal Grassmannian and a functional corresponding to the functional are derived. Using this framework a Bernstein-type theorem is formulated: If \(f:M^{n-1}\to S^ n\) \((n\neq 4,6)\) is a compact, oriented, connected Willmore hypersurface satisfying some additional condition with hyperbolic Gauß map \(\gamma_ f\) such that there exists a vector \(a\in {\mathbb{R}}_ 1^{n+2}\) with \(\langle\gamma_ f,a\rangle\neq 0\) on all \(M\), then \(f(M)\) is totally umbilic. Here the conformal Grassmannian is identified with the Lorentz sphere in Minkowski space \({\mathbb{R}}_ 1^{n+2}\), which is the parameter space of \({\mathbb{R}}P^{n+1}\).
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 510023
zbMATH Keywords: Willmore functional
zbMATH Keywords: conformal Gauß map
zbMATH Keywords: Euler-Lagrange equation
zbMATH Keywords: Darboux frame
zbMATH Keywords: Möbius space
zbMATH Keywords: conformal Grassmannian
zbMATH Keywords: totally umbilic
reviewed by: Udo Hertrich-Jeromin
MaRDI profile type: Publication
cites work: A duality theorem for Willmore surfaces
cites work: On a Variational Problem on Hypersurfaces
cites work: The conformal Gauss map of submanifolds of the Möbius space
cites work: Submanifolds of the Möbius space
cites work: A Global Invariant of Conformal Mappings in Space
cites work: Q4775119
description: scientific article; zbMATH DE number 510023